An Operator Semigroup Method for Rectangular Plates With 2 Opposite Sides Simply Supported
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摘要: 考虑弹性理论中对边简支矩形薄板方程,用算子半群方法求解问题.首先,将方程转换成抽象Cauchy问题.其次,构造空间框架并证明对应的算子矩阵生成压缩半群.最后,经Fourier变换,采用一致连续半群做逼近,进而给出对边简支矩形薄板方程的解析解.该方法自然蕴含着解的存在唯一性.
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关键词:
- 矩形板 /
- 抽象Cauchy问题 /
- C0半群 /
- 解析解
Abstract: The problem of solving a rectangular thin plate with 2 opposite sides simply supported in elasticity theory by means of the operator semigroup method was addressed. First, the plate equations were transformed into the abstract Cauchy problem. Then, the Hilbert space was defined and it was proved that the corresponding operator matrix generates contraction semigroups. Finally, the uniformly continuous semigroup approximation was applied through the Fourier transform, and the analytical solutions to the equations were given. The method naturally implies the existence and uniqueness of the solution.-
Key words:
- rectangular plate /
- abstract Cauchy problem /
- C0 semigroup /
- analytical solution
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[1] Sokolnikoff I S. Mathematical Theory of Elasticity [M]. 2nd ed. New York: McGraw-Hill Book Co Inc, 1956. [2] 钟万勰. 弹性力学求解新体系[M]. 大连: 大连理工大学出版社, 1995.(ZHONG Wan-xie. A New Systematic Methodology for Theory of Elasticity [M]. Dalian: Dalian University of Technology Press, 1995.(in Chinese)) [3] 王华, 阿拉坦仓, 黄俊杰. 弹性理论中上三角无穷维Hamilton算子根向量组的完备性[J]. 应用数学和力学, 2012,33(3): 366-378.(WANG Hua, Alatancang, HUANG Jun-jie. Completeness of the system of root vectors of upper triangular infinite-dimensional Hamiltonian operators appearing in elasticity theory[J]. Applied Mathematics and Mechanics,2012,〖STHZ〗 33(3): 366-378.(in Chinese)) [4] LI Rui, ZHONG Yang, LI Ming. Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method[J]. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences,2013,469(2153): 20120681. [5] Lim C W, Cui S, Yao W. On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported[J]. International Journal of Solids and Structures,2007,44(16): 5396-5411. [6] Eburilitu, Alatancang. Symplectic eigenfunction expansion approach for a class of rectangular plates[J]. Mathematica Applicata,2013,26(1): 80-88. [7] Lim C, Xu X. Symplectic elasticity: theory and applications[J]. Applied Mechanics Reviews,2010,63(5): 050802. [8] ZHANG Guo-ting, HUANG Jun-jie, Alatancang. Eigen-vector expansion theorem of a class of operator matrices appearing in elasticity and applications[J]. Acta Physica Sinica,2012,〖STHZ〗 61(14): 140205. [9] 侯国林, 阿拉坦仓. 对边简支的矩形平面弹性问题的辛本征展开定理[J]. 应用数学和力学, 2010,31(10): 1181-1190.(HOU Guo-lin, Alatancang. Symplectic eigenfunction expansion theorem for the rectangular plane elasticity problems with two opposite simply supported[J]. Applied Mathematics and Mechanics,2010,31(10): 1181-1190.(in Chinese)) [10] HUANG Jun-jie, Alatancang, WANG Hua. Completeness of the system of eigenvectors of off-diagonal operator matrices and its applications in elasticity theory[J]. Chinese Physics B,2010,19(12): 120201-1-120201-9. [11] Goldstein J A. Semigroups of Linear Operators and Applications [M]. New York: Oxford University Press, 1985. [12] Engel K J, Nagel R. One Parameter Semigroups for Linear Evolution Equations [M]. Graduate Texts in Mathematics 194. New York: Springer-Verlag, 2000. [13] Webster J T. Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach[J].Nonlinear Analysis,2011,74(10): 3123-3136. [14] 阿拉坦仓, 张鸿庆, 钟万勰. 矩阵多元多项式的带余除法及其应用[J]. 应用数学和力学, 2000,21(7): 661-668.(Alatancang, ZHANG Hong-qing, ZHONG Wan-xie. Pseudo-division algorithm for matrix multivariable polynomial and its application[J]. Applied Mathematics and Mechanics,2000,21(7): 661-668.(in Chinese)) [15] Kato T. Perturbation Theory for Linear Operators [M]. Classics in Mathematics. Berlin: Springer-Verlag, 1995.(reprint of the 1980 edition)
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